I assume that you are talking about governmental executives as opposed to business executives. I will answer the question using this assumption.
Immunity is a legal rule where some person cannot be prosecuted for crimes. In this case, it would mean that a governmental executive could not be prosecuted for crimes allegedly committed while he was in office.
The reason some would say this is important in developing countries is that executives would too often be prosecuted for political (as opposed to legal) reasons. In other words, it would be very hard to do anything as president of a country if you were worried that when you left office you would be prosecuted just because the next president didn't like you and your policies.
If you look at it that way, immunity for executives gives them more freedom to act without fear of someone taking revenge on them for their political decisions.
Immunity is a general word which can have may different meanings depending upon the context in which it is used. For example, a person who is unmindful of criticism of ones actions may be said to be immune to criticism. However, the word is used most frequently to mean two things. One meaning is ability to resist the effect of disease causing infections. A person having immunity against a particular disease is not likely to contract that disease in spite of being infected by the organism causing it. I do not see any justification in providing better protection only to some executives.
The second common meaning is immunity is protection from being charged and prosecuted under some law. Some degree of such immunity is often granted by countries to citizens of other countries acting in some diplomatic capacity such as in an embassy. This provision is made to ensure that in spite of state of conflict existing between countries, some degree of communication can be maintained between them without fear of the diplomats of one country being harassed or punished in another host country. This immunity also gives the diplomatic staff some additional protection for keeping secret any confidential information available to them from their home country.
Limited legal immunity may also be provided to some government official including elected politicians. The purpose of such immunity is to enable the officials to perform their duties without being hampered or harassed by legal process based on unjustified charges.
However it must be noted that such legal immunity to diplomats or government officials is justified in all countries and not just developing countries. Further, nowhere the immunity is absolute. The immunity is available to them only against some specific type of legal action. Also, there are provision for prosecution for such persons with additional checks and precautions, so that they cannot get away with really illegal acts.